Bring a large pot of salted water to a boil for the pasta.
Melt butter with olive oil in a wide skillet over medium-low heat.
Add leeks and a pinch of salt. Cook slowly, stirring, until golden and sweet, 12–15 minutes.
Stir in mushrooms. Raise heat to medium-high and brown until their juices evaporate, 6–8 minutes.
Add garlic, thyme, and red pepper flakes. Cook until fragrant, about 30 seconds.
Pour in white wine and scrape the pan. Simmer until nearly reduced, 1–2 minutes.
Stir in broth, cream, Dijon, and lemon juice. Simmer to slightly thicken, 2–3 minutes. Season lightly.
Cook pasta until just al dente. Reserve 1 cup pasta water, then drain.
Add pasta to the skillet. Take off heat. Toss in Gruyère, splashing pasta water to make it glossy.
Season with remaining salt and black pepper. Finish with Parmesan and parsley. Serve hot.